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Numbers with distinct digits in factorial base.
4

%I #21 Feb 21 2024 01:20:48

%S 0,1,2,4,5,10,13,14,19,20,22,23,46,67,68,77,82,85,86,101,106,109,110,

%T 115,116,118,119,238,355,356,461,466,469,470,503,526,547,548,557,562,

%U 565,566,623,646,667,668,677,682,685,686,701,706,709,710,715,716,718

%N Numbers with distinct digits in factorial base.

%C This sequence is a variant of A010784; however here we have infinitely many terms (for example all the terms of A033312 belong to this sequence).

%H Rémy Sigrist, <a href="/A321682/b321682.txt">Table of n, a(n) for n = 1..8179</a> (terms up to 13!)

%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>.

%e The first terms, alongside the corresponding factorial base representations, are:

%e n a(n) fac(a(n))

%e -- ---- ---------

%e 1 0 (0)

%e 2 1 (1)

%e 3 2 (1,0)

%e 4 4 (2,0)

%e 5 5 (2,1)

%e 6 10 (1,2,0)

%e 7 13 (2,0,1)

%e 8 14 (2,1,0)

%e 9 19 (3,0,1)

%e 10 20 (3,1,0)

%e 11 22 (3,2,0)

%e 12 23 (3,2,1)

%e 13 46 (1,3,2,0)

%e 14 67 (2,3,0,1)

%p b:= proc(n, i) local r; `if`(n<i, [n],

%p [b(iquo(n, i, 'r'), i+1)[], r])

%p end:

%p t:= n-> (l-> is(nops(l)=nops({l[]})))(b(n, 2)):

%p select(t, [$0..1000])[]; # _Alois P. Heinz_, Nov 16 2018

%t q[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; UnsameQ @@ s]; Select[Range[0, 720], q] (* _Amiram Eldar_, Feb 21 2024 *)

%o (PARI) is(n) = my (s=0); for (k=2, oo, if (n==0, return (1)); my (d=n%k); if (bittest(s,d), return (0), s+=2^d; n\=k))

%Y Cf. A010784, A033312, A108731.

%K nonn,base

%O 1,3

%A _Rémy Sigrist_, Nov 16 2018